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## Thursday, December 12, 2013

### fwd disc factor, fwd rate ... again

(See other posts in this blog. I think they offer simpler explanations.)

(Once we are clear on fwd disc factor, it's easy to convert it to fwd rate.)

basic idea -- discount an distant future income to tomorrow, rather than to today.

First we need to understand all the jargon around PV discounting which discounts to today...

Fwd discount factor is Discounting an income (or outflow) from a distant future date M (eg Nov) to a "nearer day" T [1] (eg Aug) is based on information available as of today "t" -- a snapshot "family photo". That discount factor could be .98. We write it as P(today, Aug, Nov) = 0.98. The fwd discount function P(t, T, M) can be interpreted as discounting \$1 income from Nov (M) to Aug (T), given information available as of today (t). Something like P( Nov -} Aug | today), reversing the order of the 3 dates. As t moves forward, more info becomes available, so we adjust our expectation and estimate to a more realistic value of .80

The core math concept is very simple once you get used to it. \$0.7 today grows to \$1 in Aug, and \$1.25 in Nov. These 2 numbers are implied/derived from today's prices. These are the risk-neutral expectations of the "growth". So \$1.25 in Nov is worth \$0.7 today, i.e.

P(Nov -} today) = 0.7/1.25. Similarly
P(Aug -} today) = 0.7/1

These are simple discount factors, Now fwd discounting is

P( Nov -} Aug | today) = 1/1.25 = 0.8

The original notation is P(today, Aug, Nov) = 0.8.

Note the 0.80 value is not discounted to today, but discounted to next month i.e. Aug only. For PV calculation, we often need to apply discounting on top of the fwd discount factor.

fwd rate is like an interest rate. 0.8 would mean 25% fwd rate.